Topic 55: Let's Talk About Sets

How do we form sets of different elements? How do those collections contribute to meaning? In this episode, we talk about set theory: the basics of how it works, how it connects to adjectives, and how it informs the way we build up larger meanings from individual words.

Quick Summary:

When looking or thinking about the world, we often group things together into collections. If we do this in a mathematical sense, we call these sets. Sets can be made out of anything - people, objects, numbers, ideas. As long as you can think of it, it's settable. And sets can be merged and separated and divided, and more. If we have one set that fits totally inside another, the smaller set is said to be a subset of the larger one. When we put two sets together, that's known as union. And when two sets overlap with each other, that section that belongs in both sets is known as an intersection.

What does this have to do with language? Well, sets underlie a lot of our semantics. If we look at adjectives, we can describe the way they behave using set theory: whether the set defined by the adjective is a subset of the full set of the noun, or only defined relative to it, or separate from the noun entirely! And that's just for starters: while the adjectives are an interesting appetizer, we can do a lot more with sets in language, and we definitely will in the future.

Extra Materials:

In the episode, we talked about some of the basic set-theoretic operations, and drew some connections to the basic arithmetic operations most of us learn about in school. So, the union of two sets is a little like adding them together, while the empty set is a bit like zero. And we can see some other parallels, too.

Like, we have set-theoretic difference, which is a bit like subtraction. As you can see below, subtracting one set from another just means taking out all the elements from the first set that you also find in the second.

{Jesse Ventura, Alex Trebek} – {Alex Trebek} = {Jesse Ventura}

We can even find something a little bit like negative numbers in set theory. When we take the complement of a set, marked with a prime (that apostrophe-looking thing), we kind of get its opposite. Specifically, we get everything outside of the set. So, if we consider a universe consisting only of people (for the sake of simplicity), then the complement of the set containing Alex Krycek would be the set of all people other than him.

{Alex Krycek}′ = {x | x is not Alex Krycek}

One of the most interesting operations, though, is the set-theoretic version of multiplication: Cartesian product. To take the Cartesian product of two sets, you form a new set that contains a bunch of small groups of elements from each of the two sets. We’ve written the result out using some abbreviations, since these things can get kind of big.

In the above set, you’ll find four smaller sets. You’ll notice that each of those smaller sets — enclosed in angled brackets — contains one element from the first set and one from the second. And these smaller sets are called ordered pairs because, unlike the original sets, the order matters. If we had done this the other way around, we'd have gotten a different result, with a different set of ordered pairs.

What makes this operation particularly interesting is what it creates. Sets of ordered pairs turn out to be something that in set theory are called relations. Relations are interesting to linguists because they’re a good way to think about the meanings of things like transitive verbs, and because they sometimes have certain mathematical properties that can end up playing a role in language as well.

Remember that, in the episode, we treated the meanings of adjectives and common nouns as sets — specifically, sets that contain all the things that those words actually apply to. So, the meaning of a word like “ghostly” would just be the set of all ghostly things, while the meaning of a noun phrase like “Men in Black” would refer to the set of all those government agents who go around covering up UFO sightings and alien encounters and the like.

“ghostly” = {x | x is ghostly}

“Men in Black” = {x | x is a Man in Black}

But transitive verbs are a bit trickier, because they always apply to two things at once — they relate things to one another. For instance, take the verb “investigated”. We could imagine that this verb is true of pairs of things, where it’s true that the first investigated the second. So, in the set that “investigated” refers to, we might find the ordered pair <Fox Mulder, the Flukeman>, or the pair <Dana Scully, Lord Kinbote>. That’s because the verb represents a relation between two sets — the set of investigators, and the set of investigatees.

Now, relations can have certain kinds of mathematical properties, and so the verbs that refer to these relations can bear those properties too. Like, a relation can be reflexive, which means that it relates everything to itself.

{<Scully, Scully>, <Skinner, Skinner>, <Mulder, Mulder>, . . . }

Verb phrases that denote reflexive relations include “is exactly as tall as” and “throws as many pencils into the ceiling as”.

A relation can also be irreflexive, which just means that it relates nothing to itself. Phrases like “is twice as tall as” or “was born before” can be said to denote irreflexive relations, because although these relations will probably contain at least some ordered pairs, they will never pair anything with itself. Even in a spooky world, you can't be born before yourself.

Relations can also be symmetric, which basically means that the relation is true in both directions, all of the time.

{<Scully, Mulder>, <Mulder, Scully>, . . . }

A verb phrase like “works with” is symmetric, since it isn’t really possible for you to work alongside someone without them also working alongside you (setting aside the complication of uncooperative coworkers). Other English examples include “played basketball with” and “lived with”.

And a relation can be asymmetric, which means it definitely only ever goes in one direction. A phrase that denotes an asymmetric relation is “has longer hair than”, since if one person has longer hair than another, the second can’t also have longer hair than the first. Other examples include “is taller than” and “runs faster than”.

Finally, relations can be transitive, which means that if one thing is related to another, and that second thing is related to a third, the first is also definitely paired with the third.

{<Bill, Samantha>, <Samantha, Fox>, <Bill, Fox>, . . . }

A phrase like “has the same last name as” signifies a transitive relation, since if two people have the same last name as each other, and one of them has the same last name as some third person, the other of those two people must also have the same last name as that third person. Other examples include “is heavier than” and “is nicer than”.

There are many other relational properties (e.g., antisymmetry, intransitivity, et cetera), and a relation can have more than one at a time, as you might have noticed. But all of this is more than just an intellectual curiosity; as if it weren’t already surprising enough that math can work as well as it does when describing language, when we conceptualize meaning in this way, we begin to uncover deeper insights into the nature of human language and how it works, as we’ll see.

We’ll definitely be looking into this stuff a lot more in the future, so stay tuned!

Discussion:

So how about it? What do you all think? Let us know below, and we’ll be happy to talk with you about sets, how we build them, and how they connect to language. There’s a lot of interesting stuff to say, and we want to hear what interests you!